which is also physically relevant [1]}, [2]}. The \(-\)  sign in the first equation in (REF ) brings considerable difficulty to investigate the persistence properties of (REF ) in the framework used in this paper because the non-local form of the the first equation in (REF ) is
\(u_t+uu_x=-\partial _x\Lambda ^{-2}\left(u^2+{2}u_x^2-{2}\overline{\rho }^2\right)\) 
